Probability density function solutions to a Bessel type pantograph equation

Abstract

We extend a cell growth model with a deterministic cell growth rate and asymmetric division to include dispersion. This extension yields a second order functional differential equation, the solution of which is required to be a probability density function for the size of a cell (as measured by DNA content). We study the equation for certain choice of non constant coeffcients that correspond to dispersion, growth and splitting rates. This choice of coeffcients leads to a Bessel type operator, and it is shown that there is a unique probability distribution function that solves the equation. The solution is constructed using the Mellin transform and is given in terms of an infinite series of modified Bessel functions.